The $C_p$-stable closure of the class of separable metrizable spaces

Taras Banakh¹ ; Saak Gabriyelyan²

¹ Ivan Franko National University of Lviv Universytetska 1 79000, Lviv, Ukraine and Institute of Mathematics Jan Kochanowski University in Kielce Świętokrzyska 15 25-406 Kielce, Poland
² Department of Mathematics Ben-Gurion University of the Negev P.O. Box 653 Be’er Sheva 84105, Israel

Colloquium Mathematicum, Tome 146 (2017) no. 2, pp. 283-294

Résumé

Denote by $\mathbf {C}_p[\mathfrak M _0]$ the $C_p$-stable closure of the class $\mathfrak M _0$ of all separable metrizable spaces, i.e., $\mathbf {C}_p[\mathfrak M _0]$ is the smallest class of topological spaces that contains $\mathfrak M _0$ and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces $C_p(X,Y)$. Using a recent deep result of Chernikov and Shelah, we prove that $\mathbf {C}_p[\mathfrak M _0]$ coincides with the class of all Tychonoff spaces of cardinality strictly less than $\beth _{\omega _1}$. Being motivated by the theory of generalized metric spaces, we also characterize other natural $C_p$-type stable closures of $\mathfrak M _0$.

Détail

DOI : 10.4064/cm6475-3-2016

Keywords: denote mathbf mathfrak p stable closure class mathfrak separable metrizable spaces mathbf mathfrak smallest class topological spaces contains mathfrak closed under taking subspaces homeomorphic images countable topological sums countable tychonoff products function spaces y using recent deep result chernikov shelah prove mathbf mathfrak coincides class tychonoff spaces cardinality strictly beth omega being motivated theory generalized metric spaces characterize other natural p type stable closures nbsp mathfrak